Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=(x-5)^{3}\)

Step-by-Step Solution

Answer

Functions are \(f(x) = x^3\) and \(g(x) = x - 5\).

Step 1: Identify the Inner Function

First, we need to determine a function that can be an inner part of a composition. The expression \((x-5)\) suggests a possibility. Let's set \(g(x) = x - 5\). This function captures the alteration happening inside the main expression.

Step 2: Define the Outer Function

With \(g(x) = x - 5\), we substitute \(g(x)\) into the expression \((x-5)^3\). The outer function can then be written as \(f(u) = u^3\), where \(u\) represents the output of \(g(x)\). Thus, \(f(g(x)) = f(x-5) = (x-5)^3\).

Step 3: Verify the Composition

To verify, substitute \(g(x) = x - 5\) into \(f(u) = u^3\) yielding \(f(g(x)) = (g(x))^3 = (x-5)^3\). This confirms that \(h(x) = f(g(x))\) using the functions we defined.

Key Concepts

Inner Function

When dealing with function composition, the first step is identifying the inner function. The inner function is like the core or the base, from which modifications and transformations of the input start. In our example where we have the function \(h(x)=(x-5)^3\), we can see that part \((x-5)\) inside the cubic expression. Here, this expression dictates what happens to the input \(x\) before any other operations take place.

• This is why we set the inner function as \(g(x) = x - 5\).
• It modifies the input \(x\) by simply subtracting 5.

By pinpointing \(g(x)\), we've isolated where the first transformation in our composite function occurs. Think of the inner function as a step that paves the way for the remaining operations.

Outer Function

Once we have the inner function in place, the next task is to define what comes after. This is where the outer function comes into play. Essentially, the outer function applies further operations to the output of the inner function, encapsulating the next layers of our operation. In the case of \((x-5)^3\), we've already isolated \(g(x) = x - 5\) as the inner function.

• The outer function is then defined by the operation the cubing, which acts on the outcome of the inner function.
• This can be written as \(f(u) = u^3\), with \(u\) representing the result of \(g(x)\).

Essentially, the outer function is the process that completes our transformation, working on the value handed off by the inner function.

Composite Functions

At the heart of function composition is the idea of combining two or more functions to achieve a more complex operation. A composite function is what we form when we apply one function to the result of another. Think of it like a sequential process where the output of one function becomes the input for the next. In the composite function \(h(x) = f(g(x))\), we have:

• \(g(x)\) as our inner function, turning \(x\) into \(x-5\).
• \(f(u) = u^3\) as our outer function, cubing the result.

The beauty of composite functions is in their flexibility and capability to express intricate transformations in mathematical problems. They allow us to break down and understand complex operations, making even the most daunting tasks easier to handle.